Linear Algebra Done Right · Problem 6A.35

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

Fix a positive integer nn. The Laplacian Δp\Delta p of a twice differentiable real-valued function pp on Rn\mathbf{R}^n is the function on Rn\mathbf{R}^n defined by Δp=∂2p∂x12+⋯+∂2p∂xn2.\Delta p = \frac{\partial^2 p}{\partial x_1^2} + \cdots + \frac{\partial^2 p}{\partial x_n^2}. The function pp is called harmonic if Δp=0\Delta p = 0. A polynomial on Rn\mathbf{R}^n is a linear combination (with coefficients in R\mathbf{R}) of functions of the form x1m1⋯xnmnx_1^{m_1} \cdots x_n^{m_n}, where m1,…,mnm_1, \ldots, m_n are nonnegative integers. Suppose qq is a polynomial on Rn\mathbf{R}^n. Prove that there exists a harmonic polynomial pp on Rn\mathbf{R}^n such that p(x)=q(x)p(x) = q(x) for every x∈Rnx \in \mathbf{R}^n with ∥x∥=1\|x\| = 1. The only fact about harmonic functions that you need for this exercise is that if pp is a harmonic function on Rn\mathbf{R}^n and p(x)=0p(x) = 0 for all x∈Rnx \in \mathbf{R}^n with ∥x∥=1\|x\| = 1, then p=0p = 0. Hint: A reasonable guess is that the desired harmonic polynomial pp is of the form q+(1−∥x∥2)rq + (1 - \|x\|^2)r for some polynomial rr. Prove that there is a polynomial rr on Rn\mathbf{R}^n such that q+(1−∥x∥2)rq + (1 - \|x\|^2)r is harmonic by defining an operator TT on a suitable vector space by Tr=Δ((1−∥x∥2)r)Tr = \Delta((1 - \|x\|^2)r) and then showing that TT is injective and hence surjective.

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